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Single Idea 10164
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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Full Idea
A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction.
Gist of Idea
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
A Reaction
This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'!
The
18 ideas
from 'Structures and Structuralism in Phil of Maths'
10166
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ZFC set theory has only 'pure' sets, without 'urelements'
[Reck/Price]
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10165
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'Analysis' is the theory of the real numbers
[Reck/Price]
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10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures
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10164
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Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
[Reck/Price]
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10168
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Formalist Structuralism says the ontology is vacuous, or formal, or inference relations
[Reck/Price]
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10172
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Set-theory gives a unified and an explicit basis for mathematics
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10171
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The existence of an infinite set is assumed by Relativist Structuralism
[Reck/Price]
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10173
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A nominalist might avoid abstract objects by just appealing to mereological sums
[Reck/Price]
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10169
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Relativist Structuralism just stipulates one successful model as its arithmetic
[Reck/Price]
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10174
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Mereological arithmetic needs infinite objects, and function definitions
[Reck/Price]
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10170
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While true-in-a-model seems relative, true-in-all-models seems not to be
[Reck/Price]
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10175
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Three types of variable in second-order logic, for objects, functions, and predicates/sets
[Reck/Price]
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10178
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Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous
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10176
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Universalist Structuralism is based on generalised if-then claims, not one particular model
[Reck/Price]
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
[Reck/Price]
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10179
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There are 'particular' structures, and 'universal' structures (what the former have in common)
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10181
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Pattern Structuralism studies what isomorphic arithmetic models have in common
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10182
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There are Formalist, Relativist, Universalist and Pattern structuralism
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